A hovercraft takes off from a platform. Its height (in meters), $x$ seconds after takeoff, is modeled by: $h(x)=-2x^2+20x+48$ What is the maximum height that the hovercraft will reach?
The hovercraft's height is modeled by a quadratic function, whose graph is a parabola. The maximum height is reached at the vertex. So in order to find the maximum height, we need to find the vertex's $y$ -coordinate. We will start by finding the vertex's $x$ -coordinate, and then plug that into $h(x)$. The vertex's $x$ -coordinate is the average of the two zeros, so let's find those first. $\begin{aligned} h(x)&=0 \\\\ -2x^2+20x+48&=0 \\\\ x^2-10x-24&=0 \\\\ (x-12)(x+2)&=0 \\\\ \swarrow &\searrow \\\\ x-12=0\text{ or }&x+2=0 \\\\ x={12}\text{ or }&x={-2} \end{aligned}$ Now let's take the zeros' average: $\dfrac{({12})+({-2})}{2}=\dfrac{10}{2}=5$ The vertex's $x$ -coordinate is $ 5$. Now let's find $h({5})$ : $\begin{aligned} h( 5)&=-2( 5)^2+20( 5)+48 \\\\ &=-50+100+48 \\\\ &=98 \end{aligned}$ In conclusion, the maximum height of the hovercraft is $98$ meters.